Q.In a large school, 80% of the pupils like mathematics. A visitor to the school asks each of 4 pupils, chosen at random, whether they like mathematics.

Let $X$ = number of pupils (out of those asked) who say 'yes'; success = 'likes mathematics' with $p=0.8$, $q=0.2$.

(i) With $n=4$: $P(0)=(0.2)^4=0.0016$; $P(1)={}^{4}C_1(0.8)(0.2)^3=0.0256$; $P(2)={}^{4}C_2(0.8)^2(0.2)^2=0.1536$; $P(3)={}^{4}C_3(0.8)^3(0.2)=0.4096$; $P(4)=(0.8)^4=0.4096$. (These five values sum to exactly 1.)

(ii)(a) With $n=4$ still: $P(X\ge2)=P(2)+P(3)+P(4)=0.1536+0.4096+0.4096=0.9728$. …

Part (i) evaluates all five p.m.f. terms of B(4,0.8); part (ii)(a) reuses the same n=4 distribution's upper tail; part (ii)(b) is a genuinely different distribution B(8,0.8), so i …

Reusing the n=4 numbers for part (ii)(b) instead of recognising that increasing the number of pupils questioned to 8 changes …